In 1925 Max Born, Werner Heisenberg, and Pascual Jordan, formulated their matrix mechanics version of "quantum mechanics" as a superior formulation of Niels Bohr's old "quantum theory." The matrix mechanics confirmed discrete states and "quantum jumps" of electrons between the energy levels, with emission or absorption of radiation. But they did not yet accept today's standard textbook view that the radiation is also discrete and in the form of Albert Einstein's spatially localized "light quanta," which were about to be renamed "photons" by American chemist Gilbert Lewis in late 1926.
In early 1926, Erwin Schrödinger developed "wave mechanics" as an alternative formulation of quantum mechanics. Schrödinger disliked the idea of discontinuous quantum jumps of discrete particles. His wave mechanics is a continuous theory, but it predicts the same energy levels and is otherwise identical to the discrete theory in its predictions. Indeed, Schrödinger proved that matrix mechanics and his wave mechanics are isomorphic theories, but that quantum mechanical calculations are much easier to do using his wave mechanics. [The author used Schrödinger's wave functions to calculate the continuous spectrum of the hydrogen quasi-molecule in his 1968 Ph.D. thesis at Harvard]
Within months of the new wave mechanics, Max Born showed that while Schrödinger's wave function evolves over time deterministically, it only predicts the positions and velocities of atomic particles statistically. Born applied to matter Einstein's view that the waves of radiation can be interpreted as probabilities for finding a light quantum. Einstein's view that the waves are "guiding fields" for the light quanta was described as public knowledge as early as 1921 by H. A. Lorentz and Louis de Broglie.
Even Heisenberg himself ultimately used Schrödinger's wave functions to calculate the "transition probabilities" for electrons to jump from one energy level to another. Schrödinger's wave mechanics is easier to visualize and much easier to calculate than Heisenberg's own matrix mechanics. Ironically, Schrödinger himself never accepted the existence of particles, neither matter nor energy, and hated the discrete "quantum jumps," preferring his continuous waves as explaining all quantum phenomena. These major disagreements between the founders of quantum mechanics continue to this day with diverse and conflicting "interpretations" of quantum mechanics, at most one of which can be correct.
In early 1927, Heisenberg announced his indeterminacy principle limiting our knowledge of the simultaneous position and velocity of atomic particles, and declared that the new quantum theory disproved causality. "We cannot - and here is where the causal law breaks down - explain why a particular atom will decay at one moment and not the next, or what causes it to emit an electron in this direction rather than that." Albert Einstein had shown this element of chance in his 1917 paper on the emission and absorption of light by matter, though Heisenberg did not explictly reference Einstein's work.
More popularly known as the Uncertainty Principle in quantum mechanics, it states that the exact position and momentum of an atomic particle can only be known within certain (sic) limits. The product of the position error Δx and the momentum error Δp is greater than or equal to Planck's constant h divided by 2π.
ΔpΔx ≥ ℏ = h/2π (1)Indeterminacy (Unbestimmtheit) was Heisenberg's original name for his principle. It is a better name than the more popular uncertainty, which connotes lack of knowledge. The Heisenberg principle is an epistemological lack of information. But it does not claim that the ontological precise position and momentum do not exist, only that we can not know them, or, as Niels Bohr put it, that we cannot say anything about them.
CausalityHeisenberg was convinced that quantum mechanics had put an end to classical ideas of causality and strict determinism. In his classic paper introducing the principle of indeterminacy, he concluded with remarks about causality.
If one assumes that the interpretation of quantum mechanics is already correct in its essential points, it may be permissible to outline briefly its consequences of principle. We have not assumed that quantum theory — in opposition to classical theory — is an essentially statistical theory in the sense that only statistical conclusions can be drawn from precise initial data. The well-known experiments of Geiger and Bothe, for example, speak directly against such an assumption. Rather, in all cases in which relations exist in classical theory between quantities which are really all exactly measurable, the corresponding exact relations also hold in quantum theory (laws of conservation of momentum and energy). possibilities and a limitation on what is possible in the future. As the statistical character of quantum theory is so closely linked to the inexactness of all perceptions, one might be led to the presumption that behind the perceived statistical world there still hides a "real" world in which causality holds. But such speculations seem to us, to say it explicitly, fruitless and senseless. Physics ought to describe only the correlation of observations. One can express the true state of affairs better in this way : Because all experiments are subject to the laws of quantum mechanics, and therefore to equation (1), it follows that quantum mechanics establishes the final failure of causality.But Heisenberg was not convinced that the lack of causality helped with the But what is wrong in the sharp formulation of the law of causality, "When we know the present precisely, we can predict the future," it is not the conclusion but the assumption that is false. Even in principle we cannot know the present in all detail. For that reason everything observed is a selection from a plenitude of problem of human freedom. He reportedly said, "We no longer have any sympathy today for the concept of 'free will'." On the other hand, his close colleague, Carl von Weizsäcker, said that Heisenberg thought about the problem of free will "all the time." ( )
On Possibilities and ActualityIn his 1955-56 Gifford Lectures Physics and Philosophy Heisenberg described a quantum mechanical probability wave as a new version of the concept of "potentia" in Aristotle's Metaphysics. He wrote,
[The] probability function follows the laws of quantum theory, and its change in the course of time, which is continuous, can be calculated from the initial conditions...The probability function combines objective and subjective elements. It contains statements about possibilities or better tendencies ("potentia” in Aristotelian philosophy), and these statements are completely objective, they do not depend on any observer; and it contains statements about our knowledge of the system, which of course are subjective in so far as they may be different for different observers.Heisenberg also refers to Aristotle's ideas on pages 121-123 and 134. "Matter is in itself not a reality but only a possibility, a "potentia"; it exists only by means of form." Unformed matter is an abstract concept. Particulate matter that "exists" has an intrinsic form, or an information structure, even if physicists have not measured it or observed it. Matter in thermal equilibrium, with maximum entropy, is said to lack information. In principle, that information exists objectively, even if we lack the subjective knowledge of it.
On Einstein's Light QuantaHeisenberg must have known that Einstein had introduced probability and causality into physics in his 1916 work on the emission and absorption of light quanta, with his explanation of transition probabilities and discovery of stimulated emission. But Heisenberg gives little credit to Einstein. In his letters to Einstein, he acknowledges that Einstein's work is relevant to indeterminacy, but does not follow through on exactly how it is relevant. And as late as the Spring of 1926, perhaps following Niels Bohr, he is not convinced of the reality of light quanta. "Whether or not I should believe in light quanta, I cannot say at this stage," he says. After Heisenberg's talk on matrix mechanics at the University of Berlin, Einstein invited him to take a walk and discuss some basic questions:
I apparently managed to arouse Einstein's interest/for he invited me to walk home with him so that we might discuss the new ideas at greater length. On the way, he asked about my studies and previous research. As soon as we were indoors, he opened the conversation with a question that bore on the philosophical background of my recent work. "What you have told us sounds extremely strange. You assume the existence of electrons inside the atom, and you are probably quite right to do so. But you refuse to consider their orbits, even though we can observe electron tracks in a cloud chamber. I should very much like to hear more about your reasons for making such strange assumptions." "We cannot observe electron orbits inside the atom," I must have replied, "but the radiation which an atom emits during discharges enables us to deduce the frequencies and corresponding amplitudes of its electrons. After all, even in the older physics wave numbers and amplitudes could be considered substitutes for electron orbits. Now, since a good theory must be based on directly observable magnitudes, I thought it more fitting to restrict myself to these, treating them, as it were, as representatives of the electron orbits." "But you don't seriously believe," Einstein protested, "that none but observable magnitudes must go into a physical theory?" "Isn't that precisely what you have done with relativity?" I asked in some surprise. "After all, you did stress the fact that it is impermissible to speak of absolute time, simply because absolute time cannot be observed; that only clock readings, be it in the moving reference system or the system at rest, are relevant to the determination of time." "Possibly I did use this kind of reasoning," Einstein admitted, "but it is nonsense all the same. Perhaps I could put it more diplomatically by saying that it may be heuristically useful to keep in mind what one has actually observed. But on principle, it is quite wrong to try founding a theory on observable magnitudes alone. In reality the very opposite happens. It is the theory which decides what we can observe. You must appreciate that observation is a very complicated process. The phenomenon under observation produces certain events in our measuring apparatus. As a result, further processes take place in the apparatus, which eventually and by complicated paths produce sense impressions and help us to fix the effects in our consciousness. Along this whole path - from the phenomenon to its fixation in our consciousness — we must be able to tell how nature functions, must know the natural laws at least in practical terms, before we can claim to have observed anything at all. Only theory, that is, knowledge of natural laws, enables us to deduce the underlying phenomena from our sense impressions. When we claim that we can observe something new, we ought really to be saying that, although we are about to formulate new natural laws that do not agree with the old ones, we nevertheless assume that the existing laws — covering the whole path from the phenomenon to our consciousness—function in such a way that we can rely upon them and hence speak of'observations'... "We shall talk about it again in a few years' time. But perhaps I may put another question to you. Quantum theory as you have expounded it in your lecture has two distinct faces. On the one hand, as Bohr himself has rightly stressed, it explains the stability of the atom; it causes the same forms to reappear time and again. On the other hand, it explains that strange discontinuity or inconstancy of nature which we observe quite clearly when we watch flashes of light on a scintillation screen. These two aspects are obviously connected. In your quantum mechanics you will have to take both into account, for instance when you speak of the emission of light by atoms. You can calculate the discrete energy values of the stationary states. Your theory can thus account for the stability of certain forms that cannot merge continuously into one another, but must differ by finite amounts and seem capable of permanent re-formation. But what happens during the emission of light?As you know, I suggested that, when an atom drops suddenly from one stationary energy value to the next, it emits the energy difference as an energy packet, a so-called light quantum. In that case, we have a particularly clear example of discontinuity. Do you think that my conception is correct? Or can you describe the transition from one stationary state to another in a more precise way?" In my reply, I must have said something like this: "Bohr has taught me that one cannot describe this process by means of the traditional concepts, i.e., as a process in time and space. With that, of course, we have said very little, no more, in fact, than that we do not know. Whether or not I should believe in light quanta, I cannot say at this stage. Radiation quite obviously involves the discontinuous elements to which you refer as light quanta. On the other hand, there is a continuous element, which appears, for instance, in interference phenomena, and which is much more simply described by the wave theory of light. But you are of course quite right to ask whether quantum mechanics has anything new to say on these terribly difficult problems. I believe that we may at least hope that it will one day. "I could, for instance, imagine that we should obtain an interesting answer if we considered the energy fluctuations of an atom during reactions with other atoms or with the radiation field. If the energy should change discontinuously, as we expect from your theory of light quanta, then the fluctuation, or, in more precise mathematical terms, the mean square fluctuation, would be greater than if the energy changed continuously. I am inclined to believe that quantum mechanics would lead to the greater value, and so establish the discontinuity. On the other hand, the continuous element, which appears in interference experiments, must also be taken into account. Perhaps one must imagine the transitions from one stationary state to the next as so many fade-outs in a film. The change is not sudden—one picture gradually fades while the next comes into focus so that, for a time, both pictures become confused and one does not know which is which. Similarly, there may well be an intermediate state in which we cannot tell whether an atom is in the upper or the lower state." "You are moving on very thin ice," Einstein warned me. "For you are suddenly speaking of what we know about nature and no longer about what nature really does. In science we ought to be concerned solely with what nature does. It might very well be that you and I know quite different things about nature. But who would be interested in that? Perhaps you and I alone. To everyone else it is a matter of complete indifference. In other words, if your theory is right, you will have to tell me sooner or later what the atom does when it passes from one stationary state to the next" "Perhaps," I may have answered. "But it seems to me that you are using language a little too strictly. Still, I do admit that everything that I might now say may sound like a cheap excuse. So let's wait and see how atomic theory develops." Einstein gave me a skeptical look. "How can you really have so much faith in your theory when so many crucial problems remain completely unsolved?"
Heisenberg's Uncertainty PrincipleWhat is the fundamental reason for our inability to measure position and momentum of a particle with arbitrary precision? Heisenberg's Microscope claims that observations "disturb" a particle's position and momentum. The higher the energy of the observing radiation (the shorter the wavelength λ), in principle the microscope has increased resolving power, allowing the particle's position to be seen more clearly. But the higher energy means that the disturbance is more severe. Max Born, Heisenberg's colleague (with Pascual Jordan) in the development of matrix mechanics, said that the resolving power of a microscope is limited simply by its aperture and the wavelength λ of the observing radiation. Born described Heisenberg's failure to pass his thesis qualifying exam in experimental physics. Heisenberg could not answer Willy Wien's qualifying exam question on the resolving power of a microscope. It was only his top grade in theoretical physics that allowed Heisenberg to get his thesis approved. Niels Bohr, Heisenberg's mentor in Copenhagen and co-developer of the Copenhagen Interpretation of quantum mechanics, showed uncertainty is not caused by a disturbance (which in principle can be accounted for). In 1927, at the Como conference, Bohr derived the uncertainty principle from the size of the minimum wave packet that can be formed from a combination of waves of different frequencies ν or wave numbers k. Heisenberg - A consequence of non-commutation of momentum p and position q operators, pq-qp ≥ h/2πi. Boltzmann's phase-space volumes realized as h3 minimal quantum volumes?
WorksTalk with Einstein (1926) Bohr-Schrödinger Meeting (1926) (PDF) Uncertainty Principle (1927) (PDF in German) Uncertainty Principle (1927) (PDF in English) History of Quantum Theory (PDF) The Copenhagen Interpretation of Quantum Theory (1955, Annotated) < The Copenhagen Interpretation of Quantum Theory (1955) (PDF)